Reconfigurable Mechanical Anisotropy in Self‐Assembled Magnetic Superstructures

Abstract Enhancement of mechanical properties in self‐assembled superstructures of magnetic nanoparticles is a new emerging aspect of their remarkable collective behavior. However, how magnetic interactions modulate mechanical properties is, to date, not fully understood. Through a comprehensive Monte Carlo investigation, this study demonstrates how the mechanical properties of self‐assembled magnetic nanocubes can be controlled intrinsically by the nanoparticle magnetocrystalline anisotropy (MA), as well as by the superstructure shape anisotropy, without any need for changes in structural design (i.e., nanoparticle size, shape, and packing arrangement). A low MA‐to‐dipolar energy ratio, as found in iron oxide and permalloy systems, favors isotropic mechanical superstructure stabilization, whereas a high ratio yields magnetically blocked nanoparticle macrospins which can give rise to metastable superferromagnetism, as expected in cobalt ferrite simple cubic supercrystals. Such full parallel alignment of the particle moments is shown to induce mechanical anisotropy, where the superior high‐strength axis can be remotely reconfigured by means of an applied magnetic field. The new concepts developed here pave the way for the experimental realization of smart magneto‐micromechanical systems (based, e.g., on the permanent super‐magnetostriction effect illustrated here) and inspire new design rules for applied functional materials.

f = fitlm(x,y,'weight',1./ey); error_slope = f.Coefficients.SE (2); error_intercept = f.Coefficients.SE(1); For cases in which a large relative error is found for the intercept value, the significance probability value of the intercept, p, was also extracted: f.Coefficients.pValue(1); A value of p larger than a typical cutoff (e.g., p > 0.05, as used here) implies that the Null Hypothesis of the intercept cannot be rejected, i.e., the data indicates that the intercept is likely to be zero. A non-linear least squares method based on a Levenberg-Marquardt algorithm was employed to obtain the data fits to an exponential function. Furthermore, the percentage increase (%increase) in cohesive from the magnetic dipolar interactions was calculated relative to the equivalent non-magnetic system of iron oxide and cobalt ferrite (i.e., the two systems become identical after removal of magnetism), by considering mean values of this nonmagnetic system (sample size 2000).

Monte Carlo Simulations
The Monte Carlo (MC) simulations were carried out by employing the Metropolis-Hastings algorithm in the canonical ensemble (NVT), [4] according to a coarse-grained interaction model established in a previous study. [3] The interactions included: (i) the van der Waals attraction between NC cores, (ii) magnetic dipole-dipole interaction between NC cores, and (iii) steric repulsion between overlapping oleic acid surfactant chains coating two approaching NCs (note that this potential did not account for any possible steric resistance to shear motion between two NCs). These are all interparticle interactions that will contribute directly to the cohesive energy, i.e., mechanical stability, of the superstructures in question. In addition, the magnetocrystalline anisotropy (MA) potential was included as an intraparticle potential dictating the orientation of the macrospin within each NC, which will have an indirect influence on the cohesive energy through the orientation-dependent magnetic dipole-dipole interactions. Macrospin orientations (steps) were sampled from a normal distributed rotational step size, averaged around zero with standard deviation of 0.01 rad. In this study, the nanoparticle positions were not fixed and thus allowed to fluctuate. Translational steps were also normal distributed: averaged around zero with standard deviation of 0.01 nm. Any possible rotations of the actual NCs were disregarded.
The bottom layer of NCs were fixed in the z-direction to pin them down on an imaginary surface (which could be any material or liquid subphase), however, no interactions between the NCs and the surface were included in the model. The simulations were performed at RT, unless stated otherwise, with RT parameters.
All simulations started with an initial rotational relaxation of 10 4 steps, in order to mimic the experimental procedure in the initial part of the simulation; nanoparticles were initially embedded in an oleic acid film which was washed off with ethanol after the magnetic field had been switched off (see the Experimental Section above). In the oleic acid film, we expected the nanoparticle positions to remain more static than in air/vacuum, and therefore we considered only rotational steps in this initial part. Following in the main part of the simulation, 100 rotational steps were performed for the macrospins of each NC in the superstructure, then followed by 100 translational steps for each NC, and then repeating this procedure until a total number of 5 • 10 5 steps was reached both rotationally and translationally. System energies were calculated and dumped after every 100 rotational and translational steps. Mean cohesive energies corresponding to the thermal equilibrium plateau ( Figure 2 in the Main Text) were calculated by averaging over the last 2000 data points of the simulation. The superstructure systems simulated at 0 K ( Figure S15 and S16 below) were relaxed at 0 K from the initial state during the course of 5 • 10 4 rotational steps. The interparticle distance was in the 0 K simulations set to that of the interaction potential minimum (occurring at one complete oleic acid chain length overlap) between two neighboring cubes and left translationally static during the simulations.
The two material systems considered in this study, namely iron oxide (Fe3O4) and cobalt ferrite (CoFe2O4), both exhibit the inverse spinel crystal structure with the same lattice constant (8.39 Å). [5] The bulk saturation magnetization of these two materials are also of similar magnitude: specifically, 4.8 • 10 5 A m -1 for iron oxide, [6] and 4.2 • 10 5 A m -1 for cobalt ferrite. [7] The magnetic moments of the NCs used in the simulations were obtained by multiplying the bulk saturation magnetizations of the respective materials with the nanoparticle volume (i.e., 12 3 nm 3 , assuming perfect cubes), followed by a multiplication of a correction factor to account for the disordered surface layer in the 12 nm cubes. [8] We used the same correction factor of 0.9 in this work as in the previous work, which was also supported by experimental magnetic measurements. [2] The MA is notably different for the two materials. In iron oxide, the (bulk) anisotropy constant was reported in the literature to be -1.3 • 10 4 J m -3 , [9][10] whereas for cobalt ferrite, different values were reported (in the range 1.8 • 10 5 -3.9 • 10 5 J m -3 ) depending on which lattice site the cobalt atom occupies in the inverse spinel crystal structure during synthesis. [11][12] In this study we chose an anisotropy constant of 2.6 • 10 5 J m -3 for cobalt ferrite.
Cobalt ferrite nanoparticles have been shown in different studies to exhibit a (2-3 times) higher anisotropy constant than the bulk value, [13][14] however, we stress that increasing the anisotropy constant in the simulations will not affect the results much, since the cobalt ferrite nanoparticles already are blocked at RT. Furthermore, the Hamaker constant, dictating the strength of the van der Waals interaction, was found in the literature to be 21.0 • 10 -20 J for magnetite (in vacuum/air). [15] The Hamaker constant of cobalt ferrite is not precisely know, [16] however, owing to the similar properties compared with iron oxide, we assume the Hamaker constant of these two materials to be equal in order to draw comparisons from the simulations solely based on their magnetic behavior. Hence, the iron oxide and cobalt ferrite materials are ideal model systems for studying the effect of magnetism on their superstructure mechanical properties.
Simulations of permalloy systems ( Figure S10 and S13 below) were performed by using the same parameters as the iron oxide systems, except for the saturation magnetization, found in the literature to be 8.6 • 10 5 A m -1 , [17] and the anisotropy constant which was set to zero. [18] The NC positions were in these simulations fixed, for a qualitative proof of concept as far as dipolar interactions are concerned. 7 Derivation of Equation (2) and (3) Derivation of Equation (1) in the Main Text has already been done in a previous work. [3] The derivation of Equation (2)

Near-Zero MA Superstructure Systems
Up to now, our results clearly demonstrate the profound effect of MA on the mechanical stability in the self-assembled superstructures. A weak (111) MA leads to a relatively disordered macrospin configuration at RT, which, in turn, enhances the mechanical stability isotropically, whereas a strong (100) MA may lead to mechanical anisotropy as a consequence of enforced parallel macrospin alignment. However, to further illustrate the role of MA in the superstructure mechanical properties, we remove the MA from the iron oxide systems (i.e., MA = 0; no intraparticle driving force influencing macrospin alignment). Interestingly, shape anisotropy becomes evident for macrospin alignment in such zero-MA iron oxide structures of high A, as shown in Figure S9 below. Specifically, in the absence of MA, the effect of shape in high A superstructures tends to align the macrospins in a super-antiferromagnetic pattern to maximize vertical head-to-tail and side-by-side antiparallel dipole-dipole interactions (reducing the demagnetizing field). Hence, shape anisotropy could impose macrospin ordering and lead to an isotropic mechanical stabilization in magnetic superstructures with a low enough MA-todipolar energy ratio. An experimental candidate for such scenario is permalloy (Ni80Fe20), with a very low MA and a saturation magnetization about twice as large as that of iron oxide (yielding magnetostatic interactions four times stronger). [17][18] In an equivalent permalloy superstructure system, thermal energy fluctuations are suppressed to a greater extent by the strong dipolar interactions, thus providing a higher degree of alignment at RT ( Figure S10).
The cohesive energy per NC is in general found to be slightly larger at RT for iron oxide with zero MA (cohesive energy plot in Figure S11), compared with the real system with non-zero MA (MA ≠ 0), as expected from the higher "freedom" of the system in the zero MA case to minimize the demagnetizing field through spin relaxation. However, for the shape anisotropy to have an effect, the aspect ratio of a superstructure should be high enough (depending on the thermal energy and the strength of the dipole-dipole interaction, e.g., A > 2 for iron oxide at RT). For A ≈ 1, no definite ordering (other than order of a short ranged and short termed nature) is observed during the entire course of the simulation for the zero MA iron oxide systems ( Figure S12), as well as in the permalloy simulations ( Figure S13). The same holds true for structures of A < 1, however in-plane macrospin alignment is favored for monolayers ( Figure   S13 and S14), which is also an obvious consequence of shape anisotropy. Figure S9. Snapshots of high A superstructures of iron oxide in thermal equilibrium, both with and without MA. A significant shape-induced vertical alignment is shown for the zero MA superstructures. As the superstructure aspect ratio reaches below 3 (i.e., for n = 5, h = 16), the effect of shape anisotropy is seen to become less profound.     In the zero MA case, a super-antiferromagnetic ordering can be observed in this snapshot, however, the direction of the alignment fluctuates between the x-and y-axis as time progresses.

Configurations
The actual macrospin configuration (via magnetic anisotropy) of the system can clearly act as another means of tailoring mechanical properties, a possibility which is non-existent in nonmagnetic systems. It is therefore important to determine the relative mechanical stability of such magnetic superstructure systems of different spin configuration. By eliminating the effect of thermal energy upon cooling to 0 K, we identify three differently ordered spin configurations resulting from the systems considered above (note that these are not necessarily the ground state configurations): SFM, super-antiferromagnetic and a spin-ice-like (i.e., super-spin-ice) phase.
In our recent work, we also showed that the iron oxide superstructures freeze into spin-ice-like patterns upon cooling to 0 K. [3] The spin-ice-like configuration results from a competition between the magnetic dipolar interactions and the MA aligning the spins close to the diagonal easy axes (effectively canceling out the demagnetizing field). In the case of the cobalt ferrite systems, the structure becomes a perfect SFM at 0 K, where the corresponding saturated demagnetizing field is sustained by the strong anisotropy along the 100 easy axes.
Furthermore, if the MA is effectively removed at 0 K, the spins align into a superantiferromagnetic pattern with flux-closure patterns at the ends of the superstructure ( Figure   S15). We compare these three different 0 K scenarios ( Figure S16) in which the spins have been relaxed at 0 K with the NC interparticle distance assumed to be one oleic acid chain length, at the interparticle potential minimum (see section above on Monte Carlo Simulations for more details). The demagnetizing field has been effectively canceled out in the spin-ice-like structure as well as in the super-antiferromagnetic structure, leading to an isotropic mechanical stabilization in both cases. In the SFM case, the structure is a perfect permanent magnet and should show a mechanical anisotropy similar to the RT case (i.e., destabilized in-plane and stabilization along the vertical direction). Overall, in terms of the cohesive energy per NC, the mechanical stability of superstructures displaying these different types of spin ordering is summarized as follows: SFM < spin-ice-like < super-antiferromagnetic (provided the magnetization, and thus the dipolar interactions, are of similar magnitude in all cases). At RT, a snapshot of the disordered superparamagnetic configuration at any given time will fall between the SFM and super-antiferromagnetic configurations (and, as previously established in the literature, will be marginally less mechanically stable than the spin-ice [3] ).  with easy axes along the 111-directions (e.g., iron oxide), the 0 K state is spin-ice-like. Strong MA of cubic symmetry along the 100-directions (e.g., cobalt ferrite) yields a metastable superferromagnetic alignment. The overall mechanical stability of superstructures displaying these types of macrospin configurations is summarized as follows: superferromagnetic < spinice-like < super-antiferromagnetic. Figure S17. The %increase in cohesive energy from the magnetic dipolar interactions calculated along the different Cartesian axes of an iron oxide supercube (n = h = 6; A = 1). As expected from a mechanically "isotropic" system, the superstructure displays values of similar magnitude along the three different axes: 53%, 57% and 56% (this small difference is expected to disappear for larger systems, or for longer simulations with averaging over a higher number of data points). Note that these values are slightly larger than the overall %increase of the same superstructure, reported in Figure 3c in the Main Text to be ≈37%. If we also take the other (low packing density) axes into account, like, e.g., the diagonals, the %increase will diminish towards this value. superstructures mimic the fibrous microstructure of natural composites, commonly found in plants and trees. One example is the familiar wooden microstructure where the strong uniaxial fibers serve to maintain the mechanical integrity of the tree, making wood a very attractive material for structural engineering. [19] Wooden microstructures can therefore inspire the design and creation of smarter artificial materials where such a high-strength axis (analogous to the wooden microfibers) could be remotely reconfigured as desired (Figure 4, Main Text), to effectively sustain loads from different directions during operation. The SEM image in the figure shows magnified wooden microfibers, in which the coordinate axes, L, R and T, denote the longitudinal, radial and tangential axes with respect to the wooden microstructure (i.e., fiber direction). The SEM image is adapted with permission. [19] Copyright 2015, Elsevier.  (3) a) The mean value is given, with the standard error in parentheses referring to the least significant digit of the corresponding value. The average interparticle spacing in the superferromagnetic cobalt ferrite superstructure is observed, as expected, to be smaller along the z-axis and larger along the x-and y-axis than that of the iron oxide system, reflecting the stronger attractive vertical and repulsive in-plane dipolar interactions (compared with superparamagnetic iron oxide with weaker isotropic attractive interactions). Furthermore, the in-plane spacings in the non-magnetic system fall in-between the values of the magnetic systems, whereas the spacing along the z-axis is observed to be larger due to the absence of magnetic attractions. Note that there is a smaller mean interparticle spacing in the isotropic systems along the z-axis relative to the x-and y-axis, which is attributed to the larger size (i.e., h > n), and thus the higher stability, along this axis, as well as the z-axis-pinning of the bottom NC layer (see the Monte Carlo Simulations section above) which suppresses the freedom of movement along this axis to some extent.